All Questions
32 questions
2
votes
1
answer
294
views
Are there always flat connections?
Let $G$ be a simply connected Lie group and $\Gamma$ a cocompact discrete and torsion-free subgroup. Is there a (real or complex) smooth vector bundle $E$ over the manifold $G/\Gamma$, which does not ...
user473423
3
votes
1
answer
349
views
Existence (or non existence) of principal bundle charts compatible with an $f$-reduction
I asked this question on math stack exchange here, but I wonder if it would be better received here.
Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, ...
- 391
4
votes
0
answers
178
views
The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
- 516
3
votes
1
answer
337
views
Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely
Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
- 516
0
votes
0
answers
84
views
The closure of the subgroup generated by a vector field may not be compact
Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group:
$$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$
where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
- 101
7
votes
1
answer
368
views
Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
- 4,081
1
vote
1
answer
345
views
Is the manifold of complex points of a quotient of compact groups just the tangent bundle?
In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
- 4,081
3
votes
2
answers
298
views
Transitive action on non-orientable $ M $ lifts to orientable double cover
Suppose that $ M $ is non-orientable with transitive action by a Lie group $ G $. Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?
This is true for ...
- 4,081
4
votes
1
answer
364
views
When is a smooth field's flow map volume preserving diffeomorphism
Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a (single) real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi_V$ defined as ...
- 43
4
votes
1
answer
304
views
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$
Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts.
1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
- 317
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
6
votes
0
answers
177
views
Equivariant Morse theory for non-compact Lie groups
Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...
- 198
3
votes
1
answer
370
views
Genericity of equivariant embeddings
I'd like to ask an equivariant version of this question.
Let $M$ be a closed manifold equipped with the action of a compact Lie group $G$. By the Mostow-Palais embedding theorem, $M$ can be embedded ...
- 1,903
8
votes
1
answer
228
views
Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at ...
- 10.5k
3
votes
1
answer
347
views
On functional equation $f\circ \exp=\exp \circ Df$ on a Riemannian manifold or a Lie Group
Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$
We search for the set $\mathcal{H}...
- 356
0
votes
0
answers
198
views
Re asking:On the proof by Chu-Kobayashi that transformation groups are Lie groups
I have similar questions asOn the proof by Chu-Kobayashi that transformation groups are Lie groups and even more, how can $Y\in\mathfrak{g}^{*}$ generate 1-parameter global transformation group of $M$ ...
2
votes
1
answer
143
views
Lie transformation group and the transformation of smooth structure from normal connected subgroup
I'm self-working on two theorems on Lie transformation group from the book Kobayashi transformation group in differential geometry, one is the following
Theorem Let $\mathfrak{S}$ the group of ...
2
votes
1
answer
208
views
Construction of $G$-invariant map between manifolds
Let $M,N$ be two closed differential manifolds and let $G$ be a compact Lie group. Assume that $G$ acts on both manifolds $M,N$ nicely (i.e. free and proper so that $M/G$ and $N/G$ have the structure ...
- 21
3
votes
0
answers
64
views
Metrically homogeneous spaces as inverse limits
Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...
- 541
15
votes
1
answer
612
views
Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?
Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
- 2,789
3
votes
0
answers
82
views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
- 6,741
2
votes
1
answer
123
views
Hamiltonian Group action with infinitely many stabiliser types
What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that ...
- 6,995
10
votes
1
answer
732
views
What's the sufficient or necessary conditions for a manifold to have Lie group structure?
For example, given a Lie group, its fundamental group must be Abelian. So $\Sigma_g$ ($g>1$) can't have Lie group structure. We also know for $S^n$ only $n=0,1,3$ can have Lie group structures.
...
- 249
5
votes
0
answers
477
views
What is the dimension of $M/G$ if it is a manifold and $G$ acts freely and smoothly?
Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have
$$\dim(M/G)=\dim M-\dim G?$$
Notes: This ...
- 779
2
votes
1
answer
120
views
$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
4
votes
1
answer
695
views
$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
4
votes
2
answers
590
views
Generalising the parametric transversality theorem to a foliation
The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all ...
- 2,099
7
votes
0
answers
516
views
Quotient of 3-sphere by binary octahedral group?
Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
- 3,059
78
votes
7
answers
8k
views
Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
- 8,559
2
votes
1
answer
571
views
On homeomorphic compact connected topological groups
I wish to thank Professor Claudio Gorodski for his very helpful
answers to my question on the webcite:
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...
- 491
14
votes
2
answers
2k
views
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?
Let $G_{1}$ and $G_{2}$ be compact connected Lie groups.
If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are
they isomorphic as Lie groups?
- 491
1
vote
2
answers
661
views
module of sections of the horizontal bundle
Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...
- 1,222